tag:blogger.com,1999:blog-25793070001740861852015-09-16T14:05:23.771-07:00Words Of MathMatt Leisshttp://www.blogger.com/profile/14363711635017311066noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-2579307000174086185.post-9898072219956811362014-04-25T17:57:00.003-07:002014-04-25T17:57:44.348-07:00"Teach Me ANYTHING in 5 Minutes" They Said:<h2>Reflections on a (Successful) Interview or How I learned to Stop Worrying and Embrace Mayo as a Metaphor for Math</h2><div>I just found out that beyond having to catch up on the #MTBoS Missions from the fall there is also now currently a 30 days of blogging challenge under the #MTBoS30 tag. Not sure I'll do an everyday blog, but that's no reason not to post now (the dishes and the 30 minutes window before I start my evening restaurant shift are, but that hopefully will excuse some of the license I'm bound to take with things like spelling, grammar, punctuation....</div><div><br /></div><div>I interviewed today for a second tutoring job. I love the place I tutor at currently, it's a centre with cats, dogs, tea and snacks, many textbooks, laptops, etc. I get emails with my appointments/cancellations, go there a bit before the appointed time and get to tutor high school preCalc, Calc, and first year Calc, plus some other stuff when it comes up.</div><div><br /></div><div>I applied for this new place because I saw an add posted online for a tutor for an elementary school student in my neighbourhood for in home tutoring. I've done this before, prior to my B.Ed program and am looking forward to how I will have changed as a tutor since then. I'm also looking at more elementary level education right now, whereas the last year or so has been more high school level. [I'm into doing it all, but have to pick jobs to apply for and the AQ/PD courses to take.]</div><div><br /></div><div>I had a skype interview that went well and I was asked to come in this morning to do a group session with a few other tutoring candidates (we were not in direct competition; I think it was to see how we engaged with other people and to save time overall). We were all asked to be prepared to teach the interviewer ANYTHING in about a 5 minute time frame, assuming no prior knowledge.&nbsp;</div><div><br /></div><div>I thought about doing the division algorithm I prefer (see previous post) or radian measures or doing some sort of lesson on math stuff. I find direct instruction kind of boring and the interviewer had said we could teach her ANYTHING and mentioned that someone had done magic, someone else had done a game; things like that. I got stuck on the idea of cooking something, in particular of making mayo.&nbsp;</div><div><br /></div><div>I have worked in restaurants for about 10 years now (on and off, mostly on, but with some time off during my undergrad and B.Ed programs; plus that year of farming that is food related but not restaurant work directly....). I worked at a French Bistro in Guelph that made handmade mayo and salad dressing exclusively. That worked out to 4-6L of each, up to 3 times a week. And by hand, I mean with a big whisk and a large bowl and me (or whomever) feeling like their arm was going to fall off. Every once and a while a new chef would use the blender and no one could tell the difference ... but the chef wanted it done by hand.</div><div><br /></div><div>Mayo is an emulsion (read: mixture) of oil and water, which don't normally like to be mixed. You can see this if you dump some oil into a glass or bowl of water. It's bound with a raw egg yolk, flavoured with vinegar and (often) mustard, salt and pepper. The only trick is to whisk REALLY hard at the start so the emulsion starts to form while adding in the oil VERY, VERY slowly.&nbsp;</div><div><br /></div><div>I think it took a bit more than 5 minutes, maybe even closer to 10. Especially considering I got the interviewer to go wash her hands (good food prep practice) while I set up the supplies on the table and poured oil into a couple 1 Cup mugs I had brought along. I had the interviewer follow along with me as we both made mayo in separate bowls.&nbsp;</div><div><br /></div><div>It got me thinking about math and cooking. How both have standard and important algorithms and practices (I actually refer to a lot of math "bits" as recipes when talking to students and think connects a bit more to their worlds). Learning one thing in both often helps learn other things. The emulsion we did is useful not just for making mayo, but also for salad dressing and sauces like hollandaise.&nbsp;</div><div><br /></div><div>In both cases though, knowing the algorithms is really far off from doing the thing. Answering a real problem with math (and mathematical thinking) and making dinner for yourself or your family are much more than factoring a quadratic or making a basic emulsion, but the more of the underlying stuff you know the easier it is for you to put it together in a way that makes sense to the context your in (and that you find pleasing to eat.) In both cases you need the confidence to think that you can do the whole thing, and that you can find your way to a solution, even if something goes off the rails somewhere along the way.&nbsp;</div><div><br /></div><div>My worlds are colliding! Now off to work.&nbsp;</div>Matt Leisshttp://www.blogger.com/profile/14363711635017311066noreply@blogger.com0tag:blogger.com,1999:blog-2579307000174086185.post-89958873602255279502014-04-23T22:50:00.001-07:002014-04-23T22:51:33.028-07:00Late to the Party<h2>Exploring the #MTBOS Mission #1: An Introduction</h2><div>Hi! I'm Matt Leiss. I'm a newly certified teacher in Toronto, Ontario. I did my undergrad in Math at McMaster University and after a few years found myself in Ottawa where I completed my B.Ed degree. I had a great experience at teacher's college and almost two years after graduating I am still excited and passionate about teaching in general and math education in particular. I'm trying out as many roles, jobs, grade levels and entry points to this career as I can. I'll try to keep my happenings posted on twitter (@mrleiss)!</div><div><br /></div><div>I want to write about what drew me into the #MTBoS and a bit about my relationship with technology. So here it goes!<br /><br />About a year ago I took a position with an online High School startup company to develop a course in line with the Ontario curriculum. My math background is/was definitely strong enough even though I took the Junior/Intermediate stream during my B.Ed (I specialized in Pure Math during my undergrad) and I had done some basic website design in high school so I knew a bit of html and knew I could find tutorials to walk me through other stuff as it came up. I didn't (and still don't) have a lot of experience creating web content, filming videos, editing photos, etc. My main experience with Learning Management Systems (LTS) was with how badly they were used during my undergrad to store course outlines and host unmoderated discussion forums.<br /><br />I had just been given an iPod Touch as a gift from my wife (I had been talking about getting one for about two years). I should note I only ever had a cell phone for about 6 months, way way before they were SMART. I have tended to take to new technology slowly, begrudgingly while rambling things about being a luddite at heart.... So I took to the podcasts, the googles, the web and eventually to the twitters. It was very slow at first. I think I added the promotional accounts for the online school, the principal, and some of the other staff and that was about it.<br /><br />My big break came when I found the podcast "On Online Education" by Eric Wignall. I listened to the episodes on the subway to and from the tutoring centre I work at and have continued to listen to them even after I stepped back from creating content for the school. (I haven't been able to find anything by him recently but if you know of other work of his or twitter contacts please post it in the comments). One episode really rocked my work. It was an interview with Maria Anderson (@busynessgirl). I have probably listened to it a half dozen times now.<br /><br />Somehow through Maria I found David Wees (@davidwees) a fellow Canadian, whose last name I assume is pronounced similar to mine (although I could be way off too!). Through these two fine folks I found some of the other rock stars of the #mtbos. Too many of you to mention; not enough of you I have dropped into my digg reader account or have been able to follow your blogs. I keep wondering if there is a way to just add all the blogs from someone else's list and then either remove or edit that as I go. Perhaps it's better to have to add blogs a few at a time so I get a better grip for what's there. I don't know?! I'm still learning how to do this internet thing!<br /><br />The Explore the Math Twitter Blog-o-Sphere Missions (<a href="http://exploremtbos.wordpress.com/">here</a>) came up while I was swamped working a contract at a private international school to teach high school math (and eventually accounting) to Mandarin speaking students from China. I wanted to take part in the challenges at the time, and I think I half did a couple of them, but I did gain a lot from lurking the posts and seeing the half conversations with other great math teachers on twitter. Just like blogs, I'm still learning and use basic twitter on my iPod to browse posts, get notifications, and favourite things I want to read later. I have tweet deck with lots of columns on my computer and am always hoping to check there more often and take part more. I added a few lists from others to my general feed and am not sure that was the best approach for me. I'm sure I'm going to go back and forth between keeping a limited scope feed and a big net approach of trying to get it all in.<br /><br />The Missions gave me the chance to find a LOT more of you, to learn about the math chats (need to get more into them...), the global math department, so many things. I also got to start to learn about you as people. Listening to Infinite Tangents by Ashli (@mythagon) and reading Justin Aion's blog (<a href="http://relearningtoteach.blogspot.com/">here</a>) have really helped me to get to know the community as a place to come for support on bad days and to feel connected to what is going on in education on a wider scale then I would know without it.<br /><br />Just like the iPod Touch, I've been talking and thinking about getting a tablet for a while now mainly because I want to read more online (mainly your blogs, but also for the AQ and PD courses I need to start taking soon) and so I can blog more on the go. My laptop battery has been fried since one of the cats chewed the cable one too many times so it's a plug in only machine. I'm hoping to go through all the challenges this spring (unless other opportunities get in the way) and blog more. Having a place to put my reflections and ideas is going to be good for me.<br /><br />My last thought for this post is that I use linux on my laptop. I think I like the challenge of having to learn how to do things, even though it can be really frustrating too. Currently I'm running Xubuntu 12.04 LTS and am having a few funny glitches, but nothing major. Geogebra and Desmos run well on it (two of the math technologies I am in shear awe of and only found because of the #mtbos) and my Dropbox and Google accounts/files seem to be fine opening in linux day, on my iPod the next, and on the windows machines I have access to at jobs.&nbsp;</div>Matt Leisshttp://www.blogger.com/profile/14363711635017311066noreply@blogger.com0tag:blogger.com,1999:blog-2579307000174086185.post-42060129396452541422014-04-18T11:18:00.001-07:002014-04-23T22:51:50.794-07:00How I'd like to teach Division<h2 style="text-align: left;">(Not Your Parent's) Long Division:</h2><div>Dividing requires finding out&nbsp;<i style="font-weight: bold;">how many "times"&nbsp;</i>a number "<i style="font-weight: bold;">goes into"</i>another. It's really just <b>counting </b>and <b>subtracting</b>.&nbsp;</div><div><br /></div><div><div style="text-align: center;">Divide 2467 by 14</div><div style="text-align: center;"><br /></div><div style="text-align: left;">Start by taking one 14 out of 2467.</div><div style="text-align: left;"><br /></div><div style="text-align: center;">2467 - 14 = 2453</div><div style="text-align: center;"><br /></div><div style="text-align: left;">And then another,</div></div><div><br /></div><div><div style="text-align: center;">2453 - 14 = 2439&nbsp;</div><div style="text-align: center;"><br /></div><div style="text-align: left;">and another!</div><div style="text-align: left;"><br /></div></div><div><div style="text-align: center;">2439 - 14 = 2425</div><div style="text-align: center;"><br /></div><div style="text-align: left;">14 has been removed 3 times. The division question, really, is asking if this pattern were continued until no more 14's could be taken out, how many 14's would have been removed in total.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Speed up by taking out ten 14's at once.</div><div style="text-align: left;"><br /></div></div><div><div style="text-align: center;">&nbsp;2439 - (14 X 10) = 2439 - 140 = 2285&nbsp;</div></div><div><br />Or 100!<br /><div style="text-align: center;">2285 - (14 X 100) = 2299 - 1400 = 885</div><div style="text-align: center;"><br /></div><div style="text-align: left;">14 has now been removed 113 times! Completing the remaining steps could look like this.</div><div style="text-align: left;"><br /></div><div style="text-align: center;">885- (10 X 14) = 855 - 140 = 745</div><div style="text-align: center;"><br /></div><div style="text-align: center;">745 - (10 X 14) = 745 - 140 = 605</div><div style="text-align: center;"><br /></div><div style="text-align: center;">605 - (10 X 14) = 605 - 140 = 465</div><div style="text-align: center;"><br /></div><div style="text-align: center;">465 - (10 X 14) = 465 - 140 = 325</div><div style="text-align: center;"><br /></div><div style="text-align: center;">325 - (10 X 14) = 325 - 140 = 185</div><div style="text-align: center;"><br /></div><div style="text-align: center;">185 - (10 X 14) = 185 - 140 = 45</div><div style="text-align: center;"><br /></div><div style="text-align: center;">45 - 14 = 31</div><div style="text-align: center;"><br /></div><div style="text-align: center;">31 - 14 = 17</div><div style="text-align: center;"><br /></div><div style="text-align: center;">17 - 14 = 3</div><div style="text-align: center;"><br /></div><div style="text-align: left;">Counting the total number of 14's removed: 3 + 10 + 100 + 60 + 3 = 176.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">That number leftover, in this case 3, is called <b>The Remainder</b>.&nbsp;</div></div><div><br /></div><div><br /></div><div><div style="text-align: center;">------------------------------</div></div><div><b><i>Reflections:</i></b></div><div><br /></div><div>Once students grasp the general mechanics of dividing numbers and can see division as a method to split into pieces, group items, etc. They are made to divide larger and larger dividend by larger and larger divisors. Ultimately students (and everyone else) reach for calculators, which is all well and good, but what about when you don't have access to a calculator (rare these days, save for the constraints of very restrictive test environments). Long division is the answer according to math teachers for the last .... years (I don't know when this algorithm came about; I'm curious).&nbsp;</div><div><br /></div><div>We shouldn't go as far as to say Long Division is bunk or useless. It's not. It is a pretty necessary algorithm used in high school division of polynomials. BUT it is overly restrictive about format and removes the choice students can make about what multiple of the divisor they want to remove at each step. Student with stronger multiplication skills may take fewer steps, but I don't see any real benefit in a student knowing that 140 goes into 1067 exactly 7 times (which is what Long Division would require here after dividing 2467 by 1 X 14 X 100).<br /><br />Long Division is a special case of this division procedure. I'm not sure which would be easier to write up a pure algorithm for or describe to a computer. Which is the kind of question I plan to put to my students in regards to algorithms and the kind of thinking I want them to develop alongside, the why does it work and when might it not work.<br /><br />I think generally it's a lot easier to remove the larger multiples first. Perhaps partly because we have more experience and therefore comfort with smaller numbers. I wonder if starting with removing a single number would slow students down in terms of seeing this.<br /><br />I think that I would plan to leave it to students to realize for themselves that they can remove multiples that are not powers of 10. I think 2 and 5 are good ones for lots of division problems.<br /><br />Oh, a game that will lead to the more traditional long division is to challenge students to the same numbers in the fewest possible steps.<br /><br />Not sure of the easiest or cleanest way to set this up on a page. The important part is that students can easily count the number of divisors removed and have space to compute the subtraction when necessary.&nbsp;</div><div><br /></div><div><br /></div>Matt Leisshttp://www.blogger.com/profile/14363711635017311066noreply@blogger.com0tag:blogger.com,1999:blog-2579307000174086185.post-61200129003860300472013-10-21T11:40:00.003-07:002014-04-23T22:51:50.797-07:00Introducing the Radian MeasureAs I've posted before, I'm teaching grade 12 Functions and am working on making better use of a great outline put together by the&nbsp;<a href="http://www.oame.on.ca/">OAME</a>. I think this is good practice for me as a new teacher because it should help me be more efficient in my lesson planning, ensure that I hit all the curriculum requirements (because this outline was built based on them), and keep me from leaning on the textbook for examples and questions. One of the skills I want to develop in my students is to help them begin to use the textbook as a reference. It will be a much better supplement if the examples and descriptions in it are not word for word what the student has already seen in class and has in her notes.<br /><div><br /></div><div>I'm feeling good about this because I think I have a BIG improvement over the lesson for introducing radians. The lesson outline really just has students think of radians as an alternate unit for angles with an easy to develop pattern that leads into an easy to use formula for converting. Just like it's better to actually spend some time discussing the differences, pros and cons, etc between the metric system and the imperial system I think it's better to start students working with radians before they even know it and have them be useful because they are connected to other areas of mathematics, then get them to find the pattern for the easy conversion formula. Hopefully I'll also remember to talk to them about how <i>any</i>&nbsp;conversion formula relies on multiplying by 1 and "cancelling" units; in this case<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-puoeCOTgI1k/UmVyLZDccPI/AAAAAAAAAAw/js6Xb3QgEVk/s1600/conversion.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-puoeCOTgI1k/UmVyLZDccPI/AAAAAAAAAAw/js6Xb3QgEVk/s1600/conversion.jpg" /></a></div><br /></div><div><br /></div><div>I'm going to use the formula for circumference and unit circles. Oh, yeah, I'm planning on writing "Matt loves circles" on the board as my hook tomorrow. If I remember I'll be bringing my pi coffee mug too. I want to lighten things a bit right after the midterm and while I also set up for student conferences where I will make sure each and every student knows how they are doing at this point and I can get some 1-1 feedback and help them set realistic goals for themselves (unfortunately some came to me pretty unprepared and the goal might end up being "learn as much as I can so I get a better grade when I take this class again"). I'm going to ask them to spend some time trying to figure out why there are 360 degrees in a circle. I'm posting that question to twitter now too to see what other responses I can get.&nbsp;</div><div><br />By using the unit circle, the circumference is 2pi and it shouldn't be a big jump to get students to tell me how far they would have to walk to get around half the circle and a quarter of the circle and 7/12ths of the circle. These being the arclengths associated with that ratio of the unit circle. I did a really quick modification to one of the included worksheets and am going to ask my students to tell me the arclength of the highlighted portion of the unit circle, as well as the angle in degrees between the terminal arm and the x-axis. (Next they'll get the formula). I tried using a highlighter at first but it didn't show up in the photocopy, so I reached for my favourite green sharpie and voila.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-A9RKKZ1j6GA/UmVzMp2thZI/AAAAAAAAAA4/7ehbpIOI0rQ/s1600/photo.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-A9RKKZ1j6GA/UmVzMp2thZI/AAAAAAAAAA4/7ehbpIOI0rQ/s1600/photo.JPG" height="239" width="320" /></a></div></div><div><br /></div><div>On a black and white photocopy the whole circle is still visible but the green is really obviously highlighted. If I keep teaching this course I might remake the page, doing something "nicer" on the computer for the same effect, but I also have some serious improvements to past portions of the course that I would want to make first. The instructions at the top of the page are not 100% accurate to what we're doing with the worksheet and in the future I will block that out to be more clear. I'll have them work on the new pairs I'm setting up for tomorrow - I've left a couple of students who need more attention at the back corner of the class for far too long already so they're going to move up and towards the middle starting tomorrow!</div><div><br /></div><div>Today was the midterm, so tomorrow my students begin our unit on Trig Functions. It should last about 2 weeks at our accelerated pace (10 hours per week of class) which works out to about 2 of the outlined lessons per day, but I plan on putting some of them together and getting into a worked example of where a trig graph might come up (I'm thinking average monthly temperature for different cities with universities in Ontario and/or Canada where my students might want to go).<br /><br />Almost forgot; the unit, lessons, and worksheets are available&nbsp;<a href="http://www.edugains.ca/resources/LearningMaterials/TIPS/tips4rm/mhf4u/Unit3_TrigonometricFunctions1.pdf">here</a>.</div>Matt Leisshttp://www.blogger.com/profile/14363711635017311066noreply@blogger.com2tag:blogger.com,1999:blog-2579307000174086185.post-86436440282654778662013-10-15T12:47:00.005-07:002014-04-23T22:51:12.003-07:00Following a PlanI guess it's normal for teachers to feel like they are starting from behind and trying to catch up. That's how I felt my first few weeks, first month, and still feel now almost half way through my short semester. For me it was a combination of being a new teacher, being hired at a school I'd never heard of right at the start of the semester, then trying to match the conflicting outlines of the other teachers who are also teaching the same course as me. My semester is also short - running from the first week of September only to the end of November.<br /><br />My teacher education program and experience at my placements did a really great job of encouraging us to not try too hard to recreate the wheel too often. There are great lessons and resources all over the place and in particular all over the web. Many of which have been created by experienced teachers, with access to time, research, and other tools that we don't have. A lot of them have even been paid to make them - others, of course, have been shared by teachers out of the goodness and kindness of their hearts and in the interest of making us all better teachers. Regardless, there seems to be no excuse for teaching from the textbook alone or for having to make up class notes and activities from scratch everyday.<br /><br />Two places we were most often directed to were the EduGAINS (mathGAINS) and TIPS4RM wesbites. My focus during my teacher education was on the intermediate grades, because I was more familiar with the high school and university material, students of that age group, and because my placements were in a grade 6 class and a grade 7/8 class. Now that I'm teaching a grade 12 course I've come back to these sites and had a closer look at the material for high school courses and I think it's just fantastic. I wish I had put together that the two sites can be seen to work in tandem but that they actual don't repeat much material. The formats for the lessons are the same and both use the titles GAINS and TIPS4RM so when I was first looking them up this Fall I thought they would have repeated the same material in two different web locations. NOT THE CASE!<br /><br />By searching for "tips4rm, mhf4u" I came across this site <a href="http://www.edu.gov.on.ca/eng/studentsuccess/lms/MHF4U.html">here</a>&nbsp;which includes a nice outline of the course and an outline for each unit along with some projects and some pedagogical explanations for why things are done the way they are. By searching for&nbsp;&nbsp;"edugains, mhf4u" I come across a link to&nbsp;<a href="http://www.edugains.ca/newsite/math2/tips4rmgrade12mhf4u.html">here</a>&nbsp;which includes the same basic outline (but not some of the projects) AND has detailed lessons for most of the topics covered. I WISH I HAD FIGURED THIS OUT SOONER! But really, I'm just glad I got myself set up on it now. Because I'm teaching the Ontario Curriculum and these were written by the Ontario Association for Math Educators (OAME) (@OAMEcounts) it's really helpful for making sure I cover all my expectations and for putting together my lesson plans in a sensible fashion (I plan on teaching this course again and as a private school I will have at least one inspection by the Ministry who will likely ask about my lesson plans, how I am tracking expectations, and things like that).<br /><br />My big idea for improving my teaching, reducing my planning time, increasing my time spent reflecting and improving (through this blog and twitter with the help of other great math teachers) is to follow these outlines as a starting point. I will modify them like crazy when I want to or need to. I have already used a great activity called "Light It Up" from the Illuminations folks too and plan on using more of their stuff so I am not going to limit myself to GAINS/TIPS4RM course either, but it means I have a great starting place that is not just teaching from the book because one of the most important skills we can develop in our (especially senior) students is the ability to use their books as resources, to use the index and glossary and look things up and then to look them up online too. That's how it works when you want to do something you don't know how to do and it's not for school. If I can teach my students how to teach themselves then I'm getting somewhere and it doesn't really matter if they ever use the math in real life....Matt Leisshttp://www.blogger.com/profile/14363711635017311066noreply@blogger.com0tag:blogger.com,1999:blog-2579307000174086185.post-28383979341893578952013-10-15T12:16:00.003-07:002014-04-23T22:51:11.999-07:00My ContextI'm a new teacher. I did a math degree and spent a few years tutoring before attending what I believe to have been a fantastic Teacher Education program in Ottawa, ON with mostly amazing faculty about a year and a half ago.After moving to Toronto I have continued to tutor and spent a good amount of time tutoring (and pre-teaching) the Functions courses (grade 11 and 12) and the Calculus and Vectors course. When I got the last minute job offer to teach high school math a private International school I jumped at the chance to teach, to put teacher on my resume and to have a paid teaching job. I'm teaching two sections of Grade 12 Functions (MHF4U under the Ontario Curriculum). I'm very comfortable with the material, with connecting one idea to the other and with cycling back to reconnect to past topics. In short I know I can and will be a good teacher. I want to be a great teacher, but I know I'm not there yet. Right now I want to be a good teacher day in and day out and I'm getting there.<br /><br />The set up where I work is different from other schools. I have small classes. I had 14 students between two sections to start and am now at 11 between the two. Unfortunately the split is 10 in one section and 1 in another. The classes can't be combined because they actually take place in different physical locations. Next week I may lose a few more students from my class of 10 as some are working well below the grade level. I wish there had been more intervention I could have done for the students who are not prepared for the class, but for a number of reasons there wasn't. I'm going to try harder at that next term and in future Septembers when there are students new to the country and new to our education system. I really like the small class size, but the class of only 1 is <i>too small</i>&nbsp;so to build in interactions with other students I'm going to connect to the class of 10 using a wikispaces page and encourage her to connect to other students through the Open Study site (<a href="http://openstudy.com/">openstudy.com</a>)<br /><br />I'm excited about being in an English Language Learning (ELL) environment. It is really great AND really challenging to realize just how high the literacy demand is for math at this level (word problems and explaining abstract concepts like limits and infinity). I'm working at reducing this demand and also at building up my students math vocabulary most every day. I have a real advantage over public school ELL situations because all of my students share a mother tongue. The international school is fed into from China and all my students speak fluent Mandarin. (So far I have failed to learn anything - I want to know the first few numbers at least by the end of term.) So our word walls have an English column and a Mandarin column and in group work I encourage the stronger English speakers to help translate and explain concepts in Mandarin to their peers. I want to keep finding ways for peer learning and instruction - another reason I'm pushing a wiki on them. If they take it up it will be a great place for them to store and share resources in Mandarin that I can't find or understand. (The only place I have yet to find anything translated from English to Mandarin and with descriptions in both is through the Khan Academy; but not much of my material is done yet and their stuff is very computational based. It's necessary, but far from sufficient for success at this level.)<br /><br />More to come!<br /><br /><br />Matt Leisshttp://www.blogger.com/profile/14363711635017311066noreply@blogger.com0tag:blogger.com,1999:blog-2579307000174086185.post-12257029762753581012013-07-15T20:08:00.000-07:002014-04-23T22:51:50.790-07:00Disambiguating the Equal Sign ("=")<h1 class="firstHeading" id="firstHeading" lang="en" style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 1.6em; font-weight: normal; line-height: 1.2em; margin: 0px 0px 0.1em; overflow: hidden; padding-bottom: 0px; padding-top: 0px;"></h1><h1 class="firstHeading" id="firstHeading" lang="en" style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 1.6em; font-weight: normal; line-height: 1.2em; margin: 0px 0px 0.1em; overflow: hidden; padding-bottom: 0px; padding-top: 0px;"><span dir="auto">Disambiguating the Equal Sign ("=")</span></h1><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">As a somewhat competent mathematician* I have been trained to view the equal sign as a firm barrier between the LS (left side) and RS (right side) of an algebraic statement and more importantly as a balance point between these two sides. Namely, whatever is on one side had better "equal" whatever is on the other side of we are fraught with mathematical (and logical) contradictions.</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">For example, the statement:&nbsp;</div><div style="font-size: medium; font-weight: normal; text-align: center;">3x + 5 = 7x + 3</div><div style="font-size: medium; font-weight: normal; text-align: center;"><br /></div><div style="font-size: medium; font-weight: normal;">&nbsp;makes mathematical "sense" in this context if and only if x has the value 1/2.&nbsp;</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">This idea of the equal sign is useful in this kind of context, where we (or more likely, our students) are being expected to solve an algebraic system; in this case a fairly simple one. It gives rise to "rules" like whatever you do to one side you have to do the other, and so on and so forth. The idea of these being to avoid the contradictions that would result in giving you a value for x that doesn't work the original statement. To get from one to the other mathematical principles are combined with logical reasoning of the "if something is true in situation A, it must also be true in situation B; because situation B can be shown to have equivilent requirements on the variables as situation A" sort.&nbsp;</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">As I noted above, it is not&nbsp;<i>us,&nbsp;</i>the mathematicians who are doing this in my head, it is our students. Meaning I am first and foremost a teacher and that mathematical context may not be shared with&nbsp;<i>them,&nbsp;</i>the students. Here is where the&nbsp;ambiguous idea of the equal sign comes in because students have different contexts, and the one context that is very important for students understanding of math that is thought of differently for mathematicians and old school mathematically trained teachers** is that of the calculator.&nbsp;</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">For many students who I have taught or tutored = is actually&nbsp;synonymous&nbsp;to the "Enter" key on a computer. It means "OK Go! Onto the next step now." After all, on a calculator you often push the = button between each mathematical step you are doing. For students in this context,</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal; text-align: center;">2 + 7 = 9 * 5 = 45 - 3 = 42 / 7 = 6</div><div style="font-size: medium; font-weight: normal; text-align: center;"><br /></div><div style="font-size: medium; font-weight: normal;">makes mathematical sense. It is even possible to follow the steps that students followed from first to last (first was to add 2 and 7, last was to divide by 7). At no point along this path does = declare that the LS and the RS are the same. Here = is shorthand for breaking up each step in the process.&nbsp;</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">As math educators I think it is important to work to understand what our students are doing and why. To look for the things they&nbsp;<i>do&nbsp;</i>get and not jump to declare something as "unclear" or "not correct." I'm not saying that this use of = is always applicable or that is superior to the traditional mathematical rigor context. In fact, I think it's important to notice that the equal sign as a barrier is important for algebra, for the manipulating of equations and solving statements; but in the context of a string of simple calculations it may actually be&nbsp;<i>easier and more clear</i>&nbsp;to simply write out what you are doing. It would save time and perhaps allow students who have trouble "showing their work" to put down what they did and in the order they did it. I can also see it being a non-trivial assignment to write a full algebriac statement for the second example.&nbsp;</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">The math conventions are there for a reason, most of them good reason; but there are situations where we have been trained to only see things when they are laid out in the fashion we are used to and to see gibberish otherwise. We need to try to be conscious of when it is useful to teach our students those conventions, when it's not, and when it's more important to see the work through their eyes.</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">How many times have you seen this type of thing on student work? What else have you previously graded as "unclear" or "incorrect" but have learned to see differently?</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">*I did my undergraduate degree in mathematics and opted for a specialist in "pure" math; which put a focus on the algebra, calculus, and analysis areas of math.</div><div style="font-size: medium; font-weight: normal;"><br /></div><div style="font-size: medium; font-weight: normal;">**I'm relatively young, but I count myself among this old guard because I (mostly) got math and found it easy to see the calculator as one tool for the&nbsp;<i>doing</i>&nbsp;of maths. In the context of this post, I also really connect with the idea of = as a hard and fast "every must be equal on both sides" idea and have corrected my students on this in the past. I have however come to see this differently when in the teacher role of trying to assess students work, students thought process, and grading&nbsp;<i>how</i>&nbsp;they got to an answer as being a correct process or not.&nbsp;</div>Matt Leisshttp://www.blogger.com/profile/14363711635017311066noreply@blogger.com0